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345edo: Difference between revisions

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Created page with "{{Infobox ET}} {{EDO intro|345}} == Theory == 345et is only consistent to the 5-limit. Using the patent val, it tempers out 2460375/2458624, 134217728/133984375, [[5120/5103]..."
 
m Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"
 
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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|345}}
{{ED intro}}


== Theory ==
== Theory ==
345et is only consistent to the 5-limit. Using the patent val, it tempers out 2460375/2458624, 134217728/133984375, [[5120/5103]], 95703125/95551488 and 359661568/358722675 in the 7-limit; 56723625/56689952, 20155392/20131375, 14348907/14348180, 1835008/1830125, 46656/46585, 2734375/2725888, 2657205/2656192, [[16384/16335]], 514714375/514434888, [[540/539]], 9765625/9732096, 369140625/369098752, [[200704/200475]], 1375/1372, 537109375/536870912, [[19712/19683]], 968000/964467, [[3025/3024]], [[532400/531441]], 202397184/201768035 and 20614528/20588575 in the 11-limit. It provides the optimal patent val for [[quartonic]] and [[kwai]].
345et is only [[consistent]] to the [[5-odd-limit]], though it has a reasonable [[13-limit]] interpretation using the [[patent val]]. It [[tempering out|tempers out]] {{monzo| 3 -18 11 }} (quartonic comma) and {{monzo| 47 -15 -10 }} (quintosec comma) in the 5-limit; [[5120/5103]], [[16875/16807]], 2460375/2458624, and 68359375/68024448 in the 7-limit; [[540/539]], 1375/1372, [[3025/3024]], [[16384/16335]], [[19712/19683]], 46656/46585, [[200704/200475]], and [[532400/531441]] in the 11-limit; and [[625/624]] and [[4225/4224]] in the 13-limit. It provides the [[optimal patent val]] for 7-limit [[kwai]].


=== Odd harmonics ===
=== Odd harmonics ===
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=== Subsets and supersets ===
=== Subsets and supersets ===
345 factors into 3 × 5 × 23, with subset edos {{EDOs|3, 5, 15, 23, 69, and 115}}.
Since 345 factors into {{factorisation|345}}, 345edo has subset edos {{EDOs| 3, 5, 15, 23, 69, and 115 }}.


== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" |[[Subgroup]]
! rowspan="2" |[[Comma list|Comma List]]
! rowspan="2" |[[Mapping]]
! rowspan="2" |Optimal<br>8ve Stretch (¢)
! colspan="2" |Tuning Error
|-
![[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
|-
|-
|2.3
! rowspan="2" | [[Subgroup]]
|{{monzo|547 -345}}
! rowspan="2" | [[Comma list]]
|{{mapping|345 547}}
! rowspan="2" | [[Mapping]]
| -0.2062
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 547 -345 }}
| {{mapping| 345 547 }}
| −0.2062
| 0.2062
| 0.2062
| 5.93
| 5.93
|-
|-
|2.3.5
| 2.3.5
|{{monzo|3 -18 11}}, {{monzo|47 -15 -10}}
| {{monzo| 3 -18 11 }}, {{monzo| 47 -15 -10 }}
|{{mapping|345 547 801}}
| {{mapping| 345 547 801 }}
| -0.1050
| −0.1050
| 0.2210
| 0.2210
| 6.35
| 6.35
|-
|-
|2.3.5.7
| 2.3.5.7
|5120/5103, 16875/16807, 95703125/95551488
| 5120/5103, 16875/16807, 68359375/68024448
|{{mapping|345 547 801 969}}
| {{mapping| 345 547 801 969 }}
| -0.2220
| −0.2220
| 0.2788
| 0.2788
| 8.02
| 8.02
|-
|-
|2.3.5.7.11
| 2.3.5.7.11
|540/539, 3025/3024, 5120/5103, 26796875/26763264
| 540/539, 1375/1372, 5120/5103, 1953125/1940598
|{{mapping|345 547 801 969 1194}}
| {{mapping| 345 547 801 969 1194 }}
| -0.2773
| −0.2773
| 0.2728
| 0.2728
| 7.84
| 7.84
|-
|-
|2.3.5.7.11.13
| 2.3.5.7.11.13
|540/539, 625/624, 3025/3024, 5120/5103, 4225/4224
| 540/539, 625/624, 1375/1372, 4225/4224, 5120/5103
|{{mapping|345 547 801 969 1194 1277}}
| {{mapping| 345 547 801 969 1194 1277 }}
| -0.2857
| −0.2857
| 0.2497
| 0.2497
| 7.18
| 7.18
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=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
! Periods<br>per 8ve
|-
! Generator<br>(reduced)*
! Periods<br />per 8ve
! Cents<br>(reduced)*
! Generator*
! Associated<br>Ratio*
! Cents*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
|1
| 1
|38\345
| 13\345
|132.17
| 45.22
|{{monzo|-38 5 13}}
| 250/243
|[[Astro]]
| [[Quartonic]] (5-limit)
|-
| 1
| 38\345
| 132.17
| {{monzo| -38 5 13 }}
| [[Astro]]
|-
|-
|1
| 1
|143\345
| 143\345
|497.39
| 497.39
|4/3
| 4/3
|[[Kwai]]
| [[Kwai]]
|-
|-
|5
| 5
|106\345<br>(32\345)
| 106\345<br />(32\345)
|368.70<br>(111.30)
| 368.70<br />(111.30)
|1024/891<br>(16/15)
| 1024/891<br />(16/15)
|[[Qintosec]]
| [[Quintosec]] (5-limit)
|}
|}
<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct


<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
[[Category:Kwai]]
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